diagram chasing in homological algebra
salamander lemma$\Rightarrow$
four lemma$\Rightarrow$ five lemma
snake lemma$\Rightarrow$ connecting homomorphism
(also nonabelian homological algebra)
The four lemma is one of the basic diagram chasing lemmas in homological algebra. It follows directly from the salamander lemma. It directly implies the five lemma.
Let $\mathcal{A}$ be an abelian category.
Consider a commuting diagram in $\mathcal{A}$ of the form
where
the rows are exact sequences,
$\tau$ is an epimorphism,
$\nu$ is a monomorphism.
Then
$\xi(ker(f)) = ker(g)$ (the image under $\xi$ of the kernel of $f$ is the kernel of $g$)
$im(f) = \eta^{-1}(im(g))$ (the preimage under $\eta$ of the image of $g$ is the image of $f$)
(the “strong four lemma”) and hence in particular
if $g$ is an epimorphism then so is $f$;
if $f$ is a monomorphism then so is $g$
(the “weak four lemma”).
A direct proof from the salamander lemma is spelled out at salamander lemma – implications – four lemma.
The strong/weak four lemma appears as lemma 3.2, 3.3 in chapter I and then with proof in lemma 3.1 of chapter XII of
Last revised on September 24, 2012 at 23:46:27. See the history of this page for a list of all contributions to it.